The colloquial Australian and New-Zealand noun dillbrain designates a foolish or silly person, an idiot, a simpleton.

This noun was derived from the colloquial Australian and New-Zealand noun dill, which designates:
– a foolish or silly person, an idiot, a simpleton;
– a victim of deception, a dupe.

In turn, the noun dill is, apparently, a back-formation from the adjective dilly, of unknown origin and meaning:
– foolish, silly;
– unable to think rationally or coherently.

The adjective dilly is first recorded (in its obsolete sense odd, eccentric) in A Glossary of Provincial Words & Phrases in Use in Somersetshire (London: Longmans, Green, Reader, & Dyer; Taunton: F. May; 1873), by Wadham Pigott Williams and William Arthur Jones [page 11]:

Dilly adj. cranky, queer.

The earliest occurrences that I have found of the noun dill are from The National Advocate (Bathurst, New South Wales, Australia) of Tuesday 2^nd May 1933 [page 2, column 2]:

NEW NAME FOR CONSTABLE

“You big dill,” said Harold Anderson, a 19 years-old [sic] Cowra youth to Constable Brown when the lights of the constable’s car shone upon Anderson as he sat beside a young lady on the running board of a car in Church Street on the night of April 27—Show night. Constable Brown objected to the name and promptly arrested Anderson, who was charged at the Police Court yesterday with having behaved in an offensive manner. Mr. J. Wilson represented defendant who did not appear. “Did you ascertain what he meant by ‘dill’?” asked Mr. Wilson after the constable had told Mr. Bromhead S.M., the facts of the case. “Dill, I presume, is short for dillbery,” ¹ interposed the Magistrate. In reply to further questions from Mr. Wilson, Constable Brown said he was in plain clothes at the time, and Anderson, who had come from Cowra with his father to bring some trotting horses to the Show, spoke more in resentment of being disturbed than anything else. Mr. Bromhead imposed a fine of 5/- in default the “rising.”

¹ I have not found the word dillbery anywhere else. The closest word that I have found is the plural noun dilberries, in the following from Norfolk Words not found in Forby’s “Vocabulary”, by W. G. Waters, published in Norfolk Archæology: Or Miscellaneous Tracts relating to the Antiquities of the County of Norfolk (Norwich: Published by Miller and Leavins for the Norfolk and Norwich Archæological Society, 1879) [page 168]:

DILBERRIES, s. Pellets of hardened dung hanging in the breech of a sheep.

The noun dill occurs on two occasions in The Australian Language (Sydney and London: Angus and Robertson Ltd., 1945), by Sidney John Baker (1912-1976):

1-: In the following list of Australian-English nouns designating “a trickster’s victim” [Chapter VII: The Underworld – page 142]:

alec, dill, gay, sim, lolly, bunny.

2-: In the following from a list of Australian-English army expressions [Chapter VIII: Australia at War – page 156]:

dill. An officer. (From the common use of dill for a simpleton or fool.)

The earliest occurrences that I have found of the noun dillbrain are as follows, in chronological order:

1-: From the following letter to the Editor, by ‘Broad Shoulders’, published in The Sydney Morning Herald (Sydney, New South Wales, Australia) of Tuesday 6^th April 1943 [No. 32,847, page 3, column 4]:

Sir,—I have been reading Mr. Curtin’s ² appeals for this Third Liberty Loan, and of the 2,000,000 people who don’t see fit to lend their share of the vital money. I might say, before I complain of these “dillbrains,” that my wife and I have been subscribing to every loan to help win this war. They say lend till it hurts; well, we are almost unconscious with pain from lending, but we are happy about it.
Too much money is held by these “dillbrains.” Some of these, I can state, are mad with money, and dangerous to our country. I watched an auction sale last week, and I wish Mr. Curtin could have been there. Let me quote some purchases by these “dopes.” A wireless set, value £52, sold for £62; cedar bed, value £6, sold for £20; a refrigerator, value £75, sold for £150. These are a few instances, but the best one was the price paid for scrap kitchenware—for one old pot, chipped and not very nice to look at, value nil, some of the smart people thought fit to bid 10/, and almost had it, when another “dillbrain” shouts 12/6 and gets it, with a smile of victory.
Just think what the amount of loose money at this particular sale could have done for our boys over there fighting for us—money wasted, because they had too much to spend. Why?

² John Curtin (1885-1945) was the 14^th Prime Minister of Australia from 1941 until his death.

2-: From the following letter to the Editor, by one Stephen C. White, published in The Sydney Morning Herald (Sydney, New South Wales, Australia) of Friday 9^th April 1943 [No. 32,850, page 3, column 5]:

Sir,—I have no sympathy with the “dillbrains” referred to by “Broad Shoulders,” but if foolish people are prepared to pay excessive prices for shoddy articles, they themselves will eventually be the losers.

3-: From 50/50 Woman’s Place (Letterettes in Fifty Words.), published in the Daily Telegraph and Daily News (Sydney, New South Wales, Australia) of Monday 14^th June 1948 [Vol. 13, No. 72, page 6, column 5]:

Custom To “Good Manners”: Ninety-nine times in 100 if a mixed pair plays off men’s tees, the woman will play first, unless she is the independent type, which plays honors as won. If each sex plays off its own tee, the woman hits off last, as the women’s tee is generally 20 or 30 yards ahead of the men’s tee.—“Dillbrain.”

4-: From The Cotton Paddock, a short story by R. Conquest, published in the World’s News (Sydney, New South Wales, Australia) of Saturday 31^st July 1948 [page 16, column 3]:

“Improvements,” Dad told us later, “are assets, an’ can be anything from cleared scrub to a line o’ fencin’. A house is an asset, so are cows—not forgettin’ a few good horses or a potato patch.”
He squinted at us. “Queensland ain’t such a greatly developed State, see, an’ that’s why these Bank blokes like t’ see us fellows improvin’ our blocks.”
“They’ve got a good City Hall in Brisbane,” young Harry said wisely. “I bet they haven’t got one like that in——”
“You’re a regular young dillbrain!” Dad said, roughly. “The City Hall doesn’t produce anything, does it? You keep your mind on the land—that’s our future, son!”